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The Work of John Tate

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The Abel Prize 2008-2012

Part of the book series: The Abel Prize ((AP))

Abstract

The article is an exposition of the work of Tate, written on the occasion of the award to him of the Abel prize. Tate’s work is explained in the context of “the great reformulation of arithmetic and geometry that has taken place since the 1950s”.

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Notes

  1. 1.

    Introduction to Tate’s talk at the conference on the Millenium Prizes (2000).

  2. 2.

    The original topology defined by Chevalley is not Hausdorff. It was Weil who pointed out the need for a topology in which the Hecke characters become the characters on J (Weil 1936). By the time of Tate’s thesis, the correct definition seems to have been common knowledge.

  3. 3.

    Let f be a Schwartz function on \(\mathbb{R}\), and let \(\hat{f}\) be its Fourier transform on \(\mathbb{\hat{R}}=\mathbb{R}\). Let ϕ be the function \(x+\mathbb{Z}\mapsto\sum_{n\in\mathbb{Z}}f(x+n)\) on \(\mathbb{R}/\mathbb {Z}\), and let \(\hat{\phi}\) be its Fourier transform on \(\widehat{\mathbb {R}/\mathbb{Z}}=\mathbb{Z}\). A direct computation shows that \(\hat{f}(n)=\hat {\phi}(n)\) for all \(n\in\mathbb{Z}\). The Fourier inversion formula says that \(\phi(x)=\sum_{n\in\mathbb{Z}}\hat{\phi}(n)\chi(x)\); in particular, \(\phi(0)=\sum_{n\in\mathbb{Z}}\hat{\phi}(n)=\sum_{n\in\mathbb {Z}}\hat {f}(n)\). But, by definition, \(\phi(0)=\sum_{n\in\mathbb {Z}}f(n)\).

  4. 4.

    See his “Notes on Artin L-functions” and the associated comments at http://publications.ias.edu/rpl/section/22.

  5. 5.

    In fact, this is not quite true, but is true for “virtual representations” with “virtual degree 0”. The decomposition of the root number of the character χ of a Galois representation is obtained by writing it as χ=(χ−dimχ⋅1)+dimχ⋅1.

  6. 6.

    Which had been discovered by Nakayama and Weil, cf. Artin and Tate, 2009, p. 189.

  7. 7.

    See Shatz, Math Reviews 0212073.

  8. 8.

    The main lacunae at the time were a rigorous intersection theory taking account of the phenomenon of pure inseparability and the construction of the Jacobian variety in nonzero characteristic.

  9. 9.

    Much has been written on these events. I’ve found the following particularly useful: Schappacher (2006).

  10. 10.

    Only the title, Valeur asymptotique du nombre des points rationnels de hauteur bornée sur une courbe elliptique, of Néron’s communication is included in the Proceedings. The sentence paraphrases one from: Lang, Serge. Les formes bilinéaires de Néron et Tate. Séminaire Bourbaki, 1963/64, Fasc. 3, Exposé 274.

  11. 11.

    Also see footnote 10.

  12. 12.

    See his Math. Reviews 0138625.

  13. 13.

    Better, it should be thought of as the pair (H,u).

  14. 14.

    Rather, this is Serre and Tate’s interpretation of what they prove; Shimura and Taniyama express their results in terms of ideals.

  15. 15.

    Katz and Tate (1999, p. 343).

  16. 16.

    “I still remember the thrill and amazement I felt when it occurred to me that the classical formulas for such an isomorphism over \(\mathbb{C}\) made sense p-adically when properly normalized.” Tate (2008).

  17. 17.

    “Tate has written to me about his elliptic curve stuff, and has asked me if I had any ideas for a global definition of analytic varieties over complete valuation fields. I must admit that I have absolutely not understood why his results might suggest the existence of such a definition, and I remain skeptical. Nor do I have the impression of having understood his theorem at all; it does nothing more than exhibit, via brute formulas, a certain isomorphism of analytic groups.” Grothendieck, letter to Serre, August 18, 1959.

  18. 18.

    “Sooner or later it will be necessary to subsume ordinary analytic spaces, rigid analytic spaces, formal schemes, and maybe even schemes themselves into a single kind of structure for which all these usual theorems will hold.” Grothendieck, letter to Serre, October 19, 1961.

  19. 19.

    As a thesis topic, Tate gave me the problem of proving a formula that he and Mike Artin had conjectured concerning algebraic surfaces over finite fields (Conjecture C below). One day he ran into me in the corridors of 2 Divinity Avenue and asked how it was going. “Not well” I said, “In one example, I computed the left hand side and got p 13; for the other side, I got p 17; 13 is not equal to 17, and so the conjecture is false.” For a moment, Tate was taken aback, but then he broke into a grin and said “That’s great! That’s really great! Mike and I must have overlooked some small factor which you have discovered.” He took me off to his office to show him. In writing it out in front of him, I discovered a mistake in my work, which in fact proved that the conjecture is correct in the example I considered. So I apologized to Tate for my carelessness. But Tate responded: “Your error was not that you made a mistake—we all make mistakes. Your error was not realizing that you must have made a mistake. This stuff is too beautiful not to be true.” Benedict Gross tells of a similar experience, but as he writes: “John was so encouraging, saying that everyone made mistakes, and the key was to understand them and to keep thinking about the problem. I felt that one of his greatest talents as an advisor was to make his students feel like we were partners in a great enterprise, modern number theory.”

  20. 20.

    Assuming the Weil conjectures, which weren’t proved until 1973.

  21. 21.

    In the literature, a number of variants of T r(V), not obviously equivalent to it, are also called the Tate conjecture. It is not always easy to discern what an author means by the “Tate conjecture”.

  22. 22.

    Since Atiyah and Hirzebruch had already found their counterexample to an integral Hodge conjecture, Tate was not tempted to state his conjecture integrally.

  23. 23.

    “One of the most exciting developments has been Elkies’ (sic) and Shioda’s construction of lattice packings from Mordell-Weil groups of elliptic curves over function fields. Such lattices have a greater density than any previously known in dimensions from about 54 to 4096.” Preface to Conway, J.H.; Sloane, N.J.A. Sphere packings, lattices and groups. Second edition. Springer-Verlag, New York, 1993.

  24. 24.

    For expository accounts, see: Carayol (2008) and Clozel (2008).

  25. 25.

    Hodge actually asked the question with \(\mathbb{Z}\)-coefficients.

  26. 26.

    Usually this is credited to Tate alone, but Tate writes: “We were both contemplating them. I think it was probably Serre who first saw clearly the simple general definition and its relation to formal groups of finite height.” The dual of a p-divisible group is often called the Serre dual.

  27. 27.

    Sen and Ax simplified and generalized Tate’s proof that C G=K, and the result is now known as the Ax-Sen-Tate theorem.

  28. 28.

    Tate 1967c, p. 180; see also Serre’s summary of Tate’s lectures (Serre 1968b, p. 324).

  29. 29.

    See Fontaine (1982) and many other articles.

  30. 30.

    “It has been widely conjectured that there is an upper bound for the rank depending only on the groundfield. This seems to me implausible because the theory makes it clear that an abelian variety can only have high rank if it is defined by equations with very large coefficients.” Cassels (1966, p. 257).

  31. 31.

    For a long time I was puzzled as to how this article came to be written, because I was not aware that Shafarevich had been allowed to travel to the West, but Tate writes: “sometime during the year 1965–1966, which I spent in Paris, Shafarevich appeared. There must have been a brief period when the Soviets relaxed their no-travel policy…. Shafarevich was in Paris for a month or so, and the paper grew out of some discussion we had. We both liked the idea of our having a joint paper, and I was happy to have it in Russian.”

  32. 32.

    Elkies, see http://web.math.hr/~duje/tors/tors.html.

  33. 33.

    About the same time, J. Blass found a more elementary proof of the same result.

  34. 34.

    Tate writes: “Early in that summer [1965], Weil had told me of the idea that all elliptic curves over \(\mathbb{Q}\) are modular [and that the conductor of the elliptic curve equals the conductor of the corresponding modular form]. That motivated Swinnerton-Dyer to make a big computer search for elliptic curves over \(\mathbb{Q}\) with not too big discriminant, in order to test Weil’s idea. But of course it was necessary to be able to compute the conductor to do that test. That was my main motivation.”

  35. 35.

    The authors assume that E is modular—at the time, it was not known that all elliptic curves over \(\mathbb{Q}\) are modular.

  36. 36.

    During a course at Princeton University; published as: Milnor (1971).

  37. 37.

    The first test of the conjecture was for \(F=\mathbb{Q}\) and i=1. Since \(\zeta_{\mathbb {Q}}(-1)=-1/12\) and \(K_{2}(\mathbb{Z})=\mathbb{Z}/2\mathbb{Z}\), the conjecture predicts that \(|K_{3}(\mathbb{Z})|\) has 24 elements, but Lee and Szczarba showed that it has 48 elements. When a seminar speaker at Harvard mentioned this, and scornfully concluded that the conjecture was false, Tate responded from the audience “Only for 2”. In fact, Lichtenbaum’s conjecture is believed to be correct up to a power of 2.

  38. 38.

    Recall that e=|μ(k)|.

  39. 39.

    Hecke’s theorem can be proved for global fields of characteristic p≠0 by methods similar to those of Hecke (Armitage 1967).

  40. 40.

    Lang calls his Chap. VII an “unpublished article of Tate”, but gives no date. In his MR review, Shatz writes that “It appears here in almost the same form the reviewer remembers from the original seminar of Tate in 1958.”

  41. 41.

    The original notes don’t give a date or a publisher. I copied this information from the footnote p. 162 of 1967a. The volume was prepared by the staff of the Institute of Advanced Study, but it was distributed by the Harvard University Mathematics Department.

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Acknowledgements

I thank B. Gross for help with dates, J.-P. Serre for correcting a misstatement, and J. Tate for answering my queries and pointing out some mistakes.

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  1. Department of Mathematics, University of Michigan, 2074 East Hall, 530 Church Street, Ann Arbor, MI, 48109-1043, USA

    J. S. Milne

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    Helge Holden

  2. Department of Mathematics, University of Oslo, Oslo, Norway

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1 Electronic Supplementary Material

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The Abel Lecture by John Tate, the Abel Laureate 2010 (MP4 269 MB)

Appendix: Bibliography of Tate’s Articles

Appendix: Bibliography of Tate’s Articles

1950s

  • 1950 Tate, John, Fourier Analysis in Number Fields and Hecke’s Zeta Functions, Ph.D. thesis, Princeton University. Published as 1967b.

  • 1951a Tate, John. On the relation between extremal points of convex sets and homomorphisms of algebras. Comm. Pure Appl. Math. 4 (1951) 31–32.

  • 1951b Artin, Emil; Tate, John T. A note on finite ring extensions. J. Math. Soc. Japan 3 (1951) 74–77.

  • 1952a Tate, John. Genus change in inseparable extensions of function fields. Proc. Amer. Math. Soc. 3 (1952) 400–406.

  • 1952b Lang, Serge; Tate, John. On Chevalley’s proof of Luroth’s theorem. Proc. Amer. Math. Soc. 3 (1952) 621–624.

  • 1952c Tate, John. The higher dimensional cohomology groups of class field theory. Ann. of Math. (2) 56 (1952) 294–297.

  • 1954 Tate, John, The Cohomology Groups of Algebraic Number Fields, pp. 66-67 in Proceedings of the International Congress of Mathematicians, Amsterdam (1954). Vol. 2. Erven P. Noordhoff N. V., Groningen; North-Holland Publishing Co., Amsterdam (1954). iv+440 pp. 19

  • 1955a Kawada, Y.; Tate, J. On the Galois cohomology of unramified extensions of function fields in one variable. Amer. J. Math. 77 (1955) 197–217.

  • 1955b Brauer, Richard; Tate, John. On the characters of finite groups. Ann. of Math. (2) 62 (1955) 1–7.

  • 1957 Tate, John. Homology of Noetherian rings and local rings. Illinois J. Math. 1 (1957) 14–27.

  • 1958a Mattuck, Arthur; Tate, John. On the inequality of Castelnuovo-Severi. Abh. Math. Sem. Univ. Hamburg 22 (1958) 295–299.

  • 1958b Tate, J. WC-groups over \(\mathfrak{p}\)-adic fields. Séminaire Bourbaki; 10e année: 1957/1958. Textes des conférences; Exposés 152 à 168; 2e éd. corrigée, Exposé 156, 13 pp. Secrétariat mathématique, Paris (1958) 189 pp (mimeographed).

  • 1958c Lang, Serge; Tate, John. Principal homogeneous spaces over abelian varieties. Amer. J. Math. 80 (1958) 659–684.

  • 1958d Tate, John, Groups of Galois Type (published as Chap. VII of Lang 1967; reprinted as Lang 1996).Footnote 40

  • 1959a Tate, John. Rational points on elliptic curves over complete fields, manuscript 1959. Published as part of 1995.

  • 1959b. Tate, John. Applications of Galois cohomology in algebraic geometry. (Written by S. Lang based on letters of Tate 1958–1959). Chap. X of: Lang, Serge. Topics in cohomology of groups. Translated from the 1967 French original by the author. Lecture Notes in Mathematics, 1625. Springer-Verlag, Berlin (1996). vi+226 pp.

1960s

  • 1961 Artin, E., and Tate., J. Class Field Theory, Harvard University, Department of Mathematics, 1961.Footnote 41 Notes from the Artin-Tate seminar on class field theory given a Princeton University 1951–1952. Reprinted as 1968c, 1990b; second edition 2009.

  • 1962a Fröhlich, A.; Serre, J.-P.; Tate, J. A different with an odd class. J. Reine Angew. Math. 209 (1962) 6–7.

  • 1962b Tate, John. Principal homogeneous spaces for Abelian varieties. J. Reine Angew. Math. 209 (1962) 98–99.

  • 1962c Tate, John, Rigid analytic spaces. Private notes, reproduced with(out) his permission by I.H.E.S (1962). Published as 1971b; Russian translation 1969a.

  • 1962d Tate, John. Duality theorems in Galois cohomology over number fields. (1963) Proc. Internat. Congr. Mathematicians (Stockholm, 1962) pp. 288–295 Inst. Mittag-Leffler, Djursholm

  • 1963 Sen, Shankar; Tate, John. Ramification groups of local fields. J. Indian Math. Soc. (N.S.) 27 (1963) 197–202 (1964).

  • 1964a Tate, John. Algebraic cohomology classes, Woods Hole 1964, 25 pages. In: Lecture notes prepared in connection with seminars held at the Summer Institute on Algebraic Geometry, Whitney Estate, Woods Hole, MA, July 6–July 31 (1964). Published as 1965b; Russian translation 1965c.

  • 1964b Tate, John (with Lubin and Serre). Elliptic curves and formal groups, 8 pages. In: Lecture notes prepared in connection with seminars held at the Summer Institute on Algebraic Geometry, Whitney Estate, Woods Hole, MA, July 6–July 31 (1964).

  • 1964c Tate, John. Nilpotent quotient groups. Topology 3 (1964) suppl. 1 109–111.

  • 1965a Lubin, Jonathan; Tate, John. Formal complex multiplication in local fields. Ann. of Math. (2) 81 (1965) 380–387.

  • 1965b Tate, John T. Algebraic cycles and poles of zeta functions. (1965) Arithmetical Algebraic Geometry (Proc. Conf. Purdue Univ., 1963) pp. 93–110 Harper & Row, New York

  • 1965c Tate, John. Algebraic cohomology classes. (Russian) Uspehi Mat. Nauk 20 (1965) no. 6 (126) 27–40.

  • 1965d Tate, John. Letter to Cassels on elliptic curve formulas. Published as 1975b.

  • 1966a Tate, John. Multiplication complexe formelle dans les corps locaux. 1966 Les Tendances Géom. en Algèbre et Théorie des Nombres pp. 257–258 Éditions du Centre National de la Recherche Scientifique, Paris.

  • 1966b Tate, John. Endomorphisms of abelian varieties over finite fields. Invent. Math. 2 (1966) 134–144.

  • 1966c Tate, J. The cohomology groups of tori in finite Galois extensions of number fields. Nagoya Math. J. 27 (1966) 709–719.

  • 1966d Lubin, Jonathan; Tate, John. Formal moduli for one-parameter formal Lie groups. Bull. Soc. Math. France 94 (1966) 49–59.

  • 1966e Tate, John T. On the conjectures of Birch and Swinnerton-Dyer and a geometric analog. 1966. Séminaire Bourbaki: Vol. 1965/66, Expose 306.

  • 1966f Tate, John. Letter to Springer, January 13 (1966). (Contains proofs of some of the theorems announced in 1962d.)

  • 1967a Tate, J. T. Global class field theory. (1967) Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965) pp. 162–203 Thompson, Washington, D.C.

  • 1967b Tate, J. T. Fourier analysis in number fields and Hecke’s zeta-functions. (1967) Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965) pp. 305–347 Thompson, Washington, D.C.

  • 1967c Tate, J. T. p-divisible groups. (1967) Proc. Conf. Local Fields (Driebergen, 1966) pp. 158–183 Springer, Berlin.

  • 1967d Tate, John. Shafarevich, I. R. The rank of elliptic curves. (Russian) Dokl. Akad. Nauk SSSR 175 (1967) 770–773.

  • 1968a Serre, Jean-Pierre; Tate, John. Good reduction of abelian varieties. Ann. of Math. (2) 88 (1968) 492–517.

  • 1968b Tate, John. Residues of differentials on curves. Ann. Sci. École Norm. Sup. (4) 1 (1968) 149–159.

  • 1968c Artin, E.; Tate, J. Class field theory. W. A. Benjamin, Inc., New York-Amsterdam (1968) xxvi+259 pp.

  • 1969a Tate, John; Rigid analytic spaces. (Russian) Mathematics: periodical collection of translations of foreign articles, Vol. 13, No. 3 (Russian), pp. 3–37. Izdat. “Mir”, Moscow (1969).

  • 1969b Tate, John, Classes d’isogénie des variétés abéliennes sur un corps fini (d’après T. Honda) Séminaire Bourbaki 352 (1968/1969).

  • 1969c Tate, John, K 2 of global fields, AMS Taped Lecture (Cambridge, Masss., Oct. 1969).

1970s

  • 1970a Tate, John; Oort, Frans. Group schemes of prime order. Ann. Sci. École Norm. Sup. (4) 3 (1970) 1–21.

  • 1970b Tate, John. Symbols in arithmetic. Actes du Congrès International des Mathèmaticiens (Nice, 1970), Tome 1, pp. 201–211. Gauthier-Villars, Paris (1971).

  • 1971 Tate, John. Rigid analytic spaces. Invent. Math. 12 (1971) 257–289.

  • 1973a Bass, H.; Tate, J. The Milnor ring of a global field. Algebraic K-theory, II: “Classical” algebraic K-theory and connections with arithmetic (Proc. Conf., Seattle, Wash., Battelle Memorial Inst., 1972), pp. 349–446. Lecture Notes in Math., Vol. 342, Springer, Berlin (1973).

  • 1973b Tate, J. Letter from Tate to Iwasawa on a relation between K 2 and Galois cohomology. Algebraic K-theory, II: “Classical” algebraic K-theory and connections with arithmetic (Proc. Conf., Seattle Res. Center, Battelle Memorial Inst., 1972), pp. 524–527. Lecture Notes in Math., Vol. 342, Springer, Berlin (1973).

  • 1973c Mazur, B.; Tate, J. Points of order 13 on elliptic curves. Invent. Math. 22 (1973/74), 41–49.

  • 1974a Tate, John T. The arithmetic of elliptic curves. Invent. Math. 23 (1974), 179–206.

  • 1974b Tate, J. The 1974 Fields medals. I. An algebraic geometer. Science 186 (1974), no. 4158, 39–40.

  • 1975a Tate, J. The work of David Mumford. Proceedings of the International Congress of Mathematicians (Vancouver, B. C., 1974), Vol. 1, pp. 11–15. Canad. Math. Congress, Montreal, Que., 1975.

  • 1975b Tate, J. Algorithm for determining the type of a singular fiber in an elliptic pencil. Modular functions of one variable, IV (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972), pp. 33–52. Lecture Notes in Math., Vol. 476, Springer, Berlin (1975).

  • 1976a Tate, J. Problem 9: The general reciprocity law. Mathematical developments arising from Hilbert problems (Proc. Sympos. Pure Math., Northern Illinois Univ., De Kalb, Ill., 1974), pp. 311–322. Proc. Sympos. Pure Math., Vol. XXVIII, Amer. Math. Soc., Providence, R.I. (1976).

  • 1976b Tate, John. Relations between K 2 and Galois cohomology. Invent. Math. 36 (1976), 257–274.

  • 1977a Tate, J. On the torsion in K 2 of fields. Algebraic number theory (Kyoto Internat. Sympos., Res. Inst. Math. Sci., Univ. Kyoto, Kyoto, 1976), pp. 243–261. Japan Soc. Promotion Sci., Tokyo (1977).

  • 1977b Tate, J. T. Local constants. Prepared in collaboration with C. J. Bushnell and M. J. Taylor. Algebraic number fields: L-functions and Galois properties (Proc. Sympos., Univ. Durham, Durham, 1975), pp. 89–131. Academic Press, London (1977).

  • 1978a Cartier, P.; Tate, J. A simple proof of the main theorem of elimination theory in algebraic geometry. Enseign. Math. (2) 24 (1978), no. 3-4, 311–317.

  • 1978b Tate, John. Fields medals. IV. Mumford, David; An instinct for the key idea. Science 202 (1978), no. 4369, 737–739.

  • 1979 Tate, J. Number theoretic background. Automorphic forms, representations and L-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 2, pp. 3–26, Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I. (1979).

1980s

  • 1981a Tate, John. On Stark’s conjectures on the behavior of L(s,χ) at s=0. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 28 (1981), no. 3, 963–978 (1982).

  • 1981b Tate, John. Brumer-Stark-Stickelberger. Seminar on Number Theory, 1980–1981 (Talence, 1980–1981), Exp. No. 24, 16 pp., Univ. Bordeaux I, Talence (1981).

  • 1981c Tate, John. On conjugation of abelian varieties of CM type. Handwritten notes (1981).

  • 1983a Tate, J. Variation of the canonical height of a point depending on a parameter. Amer. J. Math. 105 (1983), no. 1, 287–294.

  • 1983b Mazur, B.; Tate, J. Canonical height pairings via biextensions. Arithmetic and geometry, Vol. I, 195–237, Progr. Math., 35, Birkhäuser Boston, Boston, MA (1983).

  • 1983c Rosset, Shmuel; Tate, John. A reciprocity law for K 2-traces. Comment. Math. Helv. 58 (1983), no. 1, 38–47.

  • 1984 Tate, John. Les conjectures de Stark sur les fonctions L d’Artin en s=0. Notes of a course at Orsay written by Dominique Bernardi and Norbert Schappacher. Progress in Mathematics, 47. Birkhäuser Boston, Inc., Boston, MA (1984).

  • 1986 Mazur, B.; Tate, J.; Teitelbaum, J. On p-adic analogues of the conjectures of Birch and Swinnerton-Dyer. Invent. Math. 84 (1986), no. 1, 1–48.

  • 1987 Mazur, B.; Tate, J. Refined conjectures of the “Birch and Swinnerton-Dyer type”. Duke Math. J. 54 (1987), no. 2, 711–750.

  • 1989 Gross, B.; Tate, J. Commentary on algebra. A century of mathematics in America, Part II, 335–336, Hist. Math., 2, Amer. Math. Soc., Providence, RI (1989).

1990s

  • 1990a Artin, M.; Tate, J.; Van den Bergh, M. Some algebras associated to automorphisms of elliptic curves. The Grothendieck Festschrift, Vol. I, 33–85, Progr. Math., 86, Birkhäuser Boston, Boston, MA (1990).

  • 1990b Artin, Emil; Tate, John. Class field theory. Second edition. Advanced Book Classics. Addison-Wesley Publishing Company, Advanced Book Program, Redwood City, CA (1990). xxxviii+259 pp. ISBN: 0-201-51011-1

  • 1991a Artin, M.; Tate, J.; Van den Bergh, M. Modules over regular algebras of dimension 3. Invent. Math. 106 (1991), no. 2, 335–388.

  • 1991b Artin, Michael; Schelter, William; Tate, John. Quantum deformations of GLn. Comm. Pure Appl. Math. 44 (1991), no. 8-9, 879–895.

  • 1991c Mazur, B.; Tate, J. The p-adic sigma function. Duke Math. J. 62 (1991), no. 3, 663–688.

  • 1992 Silverman, Joseph H.; Tate, John. Rational points on elliptic curves. Undergraduate Texts in Mathematics. Springer-Verlag, New York (1992). x+281 pp.

  • 1994a Tate, John. Conjectures on algebraic cycles in l-adic cohomology. Motives (Seattle, WA, 1991), 71–83, Proc. Sympos. Pure Math., 55, Part 1, Amer. Math. Soc., Providence, RI (1994).

  • 1994b Tate, John. The non-existence of certain Galois extensions of \(\mathbb{Q}\) unramified outside 2. Arithmetic geometry (Tempe, AZ, 1993), 153–156, Contemp. Math., 174, Amer. Math. Soc., Providence, RI (1994).

  • 1994c Artin, Michael; Schelter, William; Tate, John. The centers of 3-dimensional Sklyanin algebras. Barsotti Symposium in Algebraic Geometry (Abano Terme, 1991), 1–10, Perspect. Math., 15, Academic Press, San Diego, CA (1994).

  • 1994d Smith, S. P.; Tate, J. The center of the 3-dimensional and 4-dimensional Sklyanin algebras. Proceedings of Conference on Algebraic Geometry and Ring Theory in honor of Michael Artin, Part I (Antwerp, 1992). K-Theory 8 (1994), no. 1, 19–63.

  • 1995 Tate, John. A review of non-Archimedean elliptic functions. Elliptic curves, modular forms, & Fermat’s last theorem (Hong Kong, 1993), 162–184, Ser. Number Theory, I, Int. Press, Cambridge, MA (1995).

  • 1996a Tate, John; Voloch, José Felipe. Linear forms in p-adic roots of unity. Internat. Math. Res. Notices (1996), no. 12, 589–601.

  • 1996b Tate, John; van den Bergh, Michel. Homological properties of Sklyanin algebras. Invent. Math. 124 (1996), no. 1-3, 619–647.

  • 1997a Tate, John. Finite flat group schemes. Modular forms and Fermat’s last theorem (Boston, MA, 1995), 121–154, Springer, New York (1997).

  • 1997b Tate, J. The work of David Mumford. Fields Medallists’ lectures, 219–223, World Sci. Ser. 20th Century Math., 5, World Sci. Publ., River Edge, NJ (1997).

  • 1999 Katz, Nicholas M.; Tate, John. Bernard Dwork (1923–1998). Notices Amer. Math. Soc. 46 (1999), no. 3, 338–343.

2000s

  • 2000 Tate, John. The millennium prize problems I, Lecture by John Tate at the Millenium Meeting of the Clay Mathematical Institute, May 2000, Paris. Video available from the CMI website.

  • 2001 Tate, John. Galois cohomology. Arithmetic algebraic geometry (Park City, UT, 1999), 465–479, IAS/Park City Math. Ser., 9, Amer. Math. Soc., Providence, RI (2001).

  • 2002 Tate, John. On a conjecture of Finotti. Bull. Braz. Math. Soc. (N.S.) 33 (2002), no. 2, 225–229.

  • 2004 Tate, John. Refining Gross’s conjecture on the values of abelian L-functions. Stark’s conjectures: recent work and new directions, 189–192, Contemp. Math., 358, Amer. Math. Soc., Providence, RI (2004).

  • 2005 Artin, Michael; Rodriguez-Villegas, Fernando; Tate, John. On the Jacobians of plane cubics. Adv. Math. 198 (2005), no. 1, 366–382.

  • 2006 Mazur, Barry; Stein, William; Tate, John. Computation of p-adic heights and log convergence. Doc. Math. (2006), Extra Vol., 577–614 (electronic).

  • 2008 Tate, John. Foreword to p-adic geometry, 9–63, Univ. Lecture Ser., 45, Amer. Math. Soc., Providence, RI (2008).

  • 2009 Artin, Emil; Tate, John. Class field theory. New edition. TeXed and slightly revised from the original 1961 version. AMS Chelsea Publishing, Providence, RI (2009). viii+194 pp.

2010s

  • 2011 Raussen, Martin; Skau, Christian. Interview with Abel Laureate John Tate. Notices Amer. Math. Soc. 58 (2011), no. 3, 444–452.

  • 2011 Tate, John. Stark’s basic conjecture. Arithmetic of L-functions, 7–31, IAS/Park City Math. Ser., 18, Amer. Math. Soc., Providence, RI.

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Milne, J.S. (2014). The Work of John Tate. In: Holden, H., Piene, R. (eds) The Abel Prize 2008-2012. The Abel Prize. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-39449-2_15

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